On Identification of Hierarchical Structure of Fuzzy Integral Systems. Katsushige FUJIMOTO Department of Architecture, Urban Planning and Building Engineering, Tohoku University, Aramaki Aoba-ku, Sendai, 980-77, Japan Phone: +81-22-217-7876, Fax: +81-22-217-7875, E-mail:[email protected] keywords:

M o¨bius inversion, IEC, finest irreducible IEC

Abstract Both Choquet and multi-linear fuzzy integral system with respect to fuzzy measure is equivalently decomposable into sub-systems using Inclusion-Exclusion Covering (IEC). Hence, IEC is a very useful concept/index. However, it is quite difficult to identify all IEC. This paper gives a method for identifying it easily, using by M o¨bius inversion(Theorem 5.1).

1

Introduction

In previous works, it is shown that InclusionExclusion Covering (IEC) is very useful and important concept for structural analysis of systems represented by fuzzy integral. Hence IEC model, that is the extended linear model using fuzzy integral model locally, is proposed and applied, with good results. However, identifying and finding out IEC is quite difficult. In fact, for objects with more than 8 attributes, it is generally impossible to identify all IEC’s by exhaustive search in real time. Recently, a method/algorithm based on GA for identifying IEC is proposed and applied, with good results. But, there is no guarantee that we will be able to identify the essential and every IEC in theoretical sense. In this paper, we characterize the IEC using M o¨bius inversion. From this characterization, we can identify the finest IEC easily. That is, we can get the essential hierarchical structure of the system represented by fuzzy integral easily. Above results is obtained by Theorem 5.1, it is the main theorem in this paper.

2

Basic Definitions

Throughout this paper, we assume that X is a finite set and X is 2X .

Definition 2.1 (fuzzy measure) A non-monotonic fuzzy measure on (X, X ) is a real valued set function µ : X → R satisfying µ(∅) = 0. Definition 2.2 (Choquet integral) For every measurable function f on X, can be written as a simple function f=

n X

(ai − ai−1 )1Ai +

m X

(bi − bi−1 )1Bi ,

(1)

i=1

i=1

where bm < · · · < b2 < b1 < b0 = 0 = a0 < a1 < a2 < · · · < an and (X \ Bm ) ⊃ · · · ⊃ (X \ B1 ) ⊃ A1 · · · ⊃ An . Choquet integral of f , written as (1), with respect to µ is defined by Z n X (ai − ai−1 )µ(Ai ) (C) f dµ = i=1

+

m X

(bi − bi−1 ){µ(X) − µ(X \ Bi )}.

i=1

(2) That is, from the fact that µ is finite, Z Z (C) f dµ = (C) (f − bm )dµ + bm µ(X).

(3)

Definition 2.3 (M o¨bius inversion) The M o¨bius inversion µM (A) of non-monotonic fuzzy measure µ on (X, X ) with respect to A ∈ X is defined by X µM (A) ≡ (−1)|A\B| µ(B). (4) B⊆A

If µ is additive, M

µ (A) =



µ(A) 0

if if

|A| = 1, otherwise.

(5)

Definition 2.4 (multi-linear fuzzy integral) The multi-linear fuzzy integral of a measurable function f : X → R with respect to a non-monotonic fuzzy measure µ is defined by Z X Y (M L) f dµ ≡ ( f (x)) · µM (B), (6) A

B⊆A x∈B

Lemma 3.1 [4] Let C = {Ci }i∈{1,...,n} be a measurable covering of X. Then there exists an n-ary class of nonmonotonic fuzzy measures {µCi (·)}i∈{1,...,n} on {A ∩ Ci |A ∈ X }i∈{1,...,n} such that X µ(A) = (11) µCi (A ∩ Ci ), ∀A ∈ X , i∈{1,...,n}

where µM is the M o¨bius inversion of non-monotonic fuzzy measure µ. Clearly, if µ is additive, from definition2.3, this coincides with the ordinary Lebesgue integral. Definition 2.5 (null set) A set N ∈ X is called a null set (with respect to µ) if µ(A ∪ N ) = µ(A) (7)

if and only if C is an IEC. Proposition 3.2 Choquet integral of a measurable function f :X → R+ with respect to a non-monotonic fuzzy measure µ is also represented using the M o¨bius inversion as follows: Z X ^ (C) f dµ ≡ ( f (x)) · µM (A), (12) X

A⊆X x∈A

for every A ∈ X .

where

Definition 2.6 (IEC) A finite measurable covering {Ci }i∈{1,···,n} of X is called an Inclusion-Exclusion Covering of X (with respect to µ) if X \ µ(A) = (−1)|I|+1 µ( Ci ∩ A) (8)

Thus, the Choquet integral also can be interpreted as the sum of the product of a kind of values/evaluations ∧x∈A f (x) on A and a kind of weights µM (A) of A for every A ∈ X . That Q is, Choquet integral is obtained by replacing x∈A f (x) of multilinear fuzzy integral by ∧x∈A f (x).

4

for every A ∈ X . If µ is additive, it follows from the “principle of inclusion and exclusion” that every measurable covering of X is an IEC.

Preliminary Propositions

Proposition 3.1 [1] If N is a null set, then Z Z (C) f dµ = (C) f dµ X

and (M L)

Z

f (x) = minx∈A f (x).

Z

f dµ

(10)

X\N

for every measurable function f on X. The above proposition shows that a null set (of attributes) is a set (of attributes) which does not influence overall evaluation. Hence by removing such a set (of attributes) we can get a model which is simpler than, and equivalent to, the original one. Considering hierarchical decomposition of the system represented by Choquet or multi-linear fuzzy integral, the following lemma will become very useful and powerful tool.

Hierarchical Decomposition

In this section, we give a hierarchical decomposition theorem for the Choquet and the multi-linear fuzzy integral, and discuss the structural analysis method based on the theorem for both the models. We use the term “the fuzzy integral” as “both the Choquet and the multi-linear fuzzy integral,” and the fuzzy integral of a measurable function f with respect to µ R is denoted by (F ) f dµ.

4.1

(9)

X\N

f dµ = (M L) X

x∈A

i∈I

I⊆{1,···,n} I6=∅

3

V

Theorems

Lemma 4.1 (M o¨bius inversion formula) [5] Let µM be the M o¨bius inversion of µ. Then, X µ(A) = µM (B). (13) B⊆A

In reverse, if equation(13) holds then µM is the M o¨bius inversion of µ. Theorem 4.1 [4] Let C = {Ci }i∈{1,...,n} be a measurable covering of X. Then there exists an n-ary class of nonmonotonic fuzzy measures {µCi (·)}i∈{1,...,n} on {A ∩ Ci |A ∈ X }i∈{1,...,n} such that Z Z X (C) f dµ = (C) f dµCi X

i∈{1,...,n}

Ci

and (M L)

Z

X

f dµ = X

(M L)

i∈{1,...,n}

Z

f dµCi

Proposition 4.1 [4] Let C and D be two finite measurable coverings of X. If C is an IEC and C ⊑ D, then D is also an IEC.

Ci

for every measurable function f on X, if and only if C is an IEC.

Proposition 4.2 [4] Let C be an IEC. If there exist C, D ∈ C such that C ⊂ 6= D then C − {C} is also an IEC.

That is, if there exists an IEC, ordinary fuzzy integral model is equivalently decomposable into hierarchical one as shown in figure 1.

Hence all we have to do is focus on the irreducible IEC which is defined as follows.

x1

Definition 4.2 (irreducible covering) An irreducible covering C of X is a finite covering satisfying the following:

f (F) f dµC C1

1

x2 x3 x4

C = D, whenever C ⊆ D, C, D ∈ C. (F) f dµC C2

(F) f dµ / C C3

Proposition 4.3 [4] Let C and D be two IEC’s. Then C ⊓ D is also an IEC.

2

3

Σ

z

x5 (F) f dµC4

x6

C4

Figure 1: IECmodel

4.2

Proposition 4.4 [4] For each finite measurable covering C of X, there exists a finite unique measurable irreducible covering D such that C ⊑ D and D ⊑ C. This D is called the irreducible covering of C Theorem 4.2 [4] There exists a unique irreducible IEC C such that C ⊑ D for every IEC D. This C is called the finest irreducible IEC and is obtained as the inter-refinement of all IEC’s.

Structure on the set of IEC’s

In previous subsection, we show that the IEC is a very useful tool/index for structural analysis of the system represented by fuzzy integral. However, there exist generally more than one IEC, that is, we obtain more than one structure of it. Then, how do we grasp the structure of that model? In this subsection, we show some properties on the set of all IEC’s for obtaining the essential structure of the system represented by fuzzy integral. Definition 4.1 Let C and D be two finite measurable coverings of X. If for every D ∈ D, there exists C ∈ C such that D ⊆ C, we write D ⊑ C. As the important operation on the set of all finite measurable coverings, we introduce the inter-refinement of C and D is denoted by C ⊓ D, and defined by C ⊓ D ≡ {C ∩ D | C ∈ C, D ∈ D} The relation ⊑ is clearly reflexive and transitive. For two measurable coverings C and D, C⊓D is clearly a measurable covering, and clearly C ⊓ D ⊑ C, D

From Proposition 4.1 and Theorem 4.2, we can see that every IEC is generated by the finest irreducible IEC. That is, every structure obtained by an IEC is generated by the structure obtained by the finest irreducible IEC. Hence, all we have to do is focus on and identify the finest irreducible IEC.

C

x1

x2

x3

x4

D

x1

x2

x3

x4

C D

x1

x2

x3

x4

irreduciable coverring of C D

x1

x2

x3

x4

Figure 2: The set of IEC’s

5

M o¨bius Inversion and IEC

In previous section, we give an interpretation and a role of the finest irreducible IEC on considering the hierarchical decomposition of systems represented fuzzy integral. The IEC is a very useful tool/index for a structural analysis and a hierarchical decomposition of that. However, as described in Introduction, the identification of finest irreducible IEC is quite difficult. Because as the number of attributes increase as the number of all measurable coverings increase astronomically, it is impossible to examining whether each measurable covering is IEC of X or not. In this section, we characterize IEC using M o¨bius inversion. By this result, we can identify the finest irreducible IEC easily. Lemma 5.1 Let C = {Ci }i∈{1,...,n} be an IEC of X with respect to µ. Then µM (A) = 0 for every A ∈ X such that A 6⊆ Ci ∀i ∈ {1, . . . , n}.

C1 A

C2

not influence the overall evaluation at all is removed beforehand in advance. Theorem 5.1 The irreducible covering C of {A ∈ X |µM (A) 6= 0} is the finest irreducible IEC of X with respect to µ. The IEC is characterized by the following corollary. Corollary 5.2 Let C be a measurable covering of X. Then C is an IEC, if and only if {A ∈ X |µM (A) 6= 0} ⊑ C.

6

Conclusions

When dealing with real problems, the fuzzy integral model with strictly 2n −1 parameters is not always requiered. Hence, IEC model, that is situating between the linear model and strict fuzzy integral model, is expected to be useful model. And so, we can see from the results with respect to “the relation between IEC and M o¨bius inversion” obtained in this paper that M o¨bius inversion is a very powerful index/tool for interpretation of the fuzzy measure and for identifying the IEC.

References

C3

Figure 3: IEC and M o¨bius inversion From above propositions, Qif {C1 , C2 , C3 } is IEC, V the evaluation ( x∈A f (x), x∈A f (x)) of a subject f on the set of attributes A as shown in figure 3 does not influence overall evaluation. That is, the set of attribute A which influence overall evaluation is only the sets included in each Ci . This results is compatible with the fact that the system represented by fuzzy integral is decomposable into the hierarchical one by an IEC (Theorem 4.1). Corollary 5.1 If there exists a non-empty null set N ∈ X , then µM (A) = 0 for every A ∈ X such that N ⊆ A.

Assumption: Now, we assume that “A ∈ X is the empty set whenever A is a null set”. This assumption means that the set of attributes which does

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